Questions tagged [convexity]

For questions involving the concept of convexity

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Convexification of difference of convex functions

I am looking for a reference/a hint to the following problem: We are given $f_1(x),f_2(x)$ convex functions (say, on $\mathbb R^d$) such that $f_1(x) \to\infty$ for $\|x\|\to\infty$. Also there is an $...
Philipp Wacker's user avatar
19 votes
4 answers
3k views

Strange result about convexity

$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$. Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ? Source: AoPS
Dattier's user avatar
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Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. ...
ccriscitiello's user avatar
6 votes
0 answers
192 views

Does the ball maximize the "kissing probability" of symmetric convex bodies? [duplicate]

Given a symmetric convex body $K \subset \mathbb{R}^n$ (i.e., a bounded symmetric convex set with non-empty interior), I am interested in the following quantity $$p_K := \Pr_{x_1, x_2 \sim K}[x_1 \in ...
Noah Stephens-Davidowitz's user avatar
1 vote
0 answers
112 views

How to prove the inequality $\ln\frac{1+e^{-y}}{1+e^{-x}}+\frac{1}{1+e^x}(y-x)\geq 0$?

I am trying to prove that $\ell(\beta) = \sum_{i=1}^n \left (-y_i \beta^{\top}x_i + \ln \left (1 + e^{\beta^{\top}x_i }\right )\right )$ is a convex function. I follow the following steps: Let $\...
Daniel Liu's user avatar
4 votes
2 answers
969 views

Convexity of $(X, y) \mapsto y^T X^{-1} y$ [closed]

Let $y \in \mathbb{R}^n$, $X \in \mathcal{S}^n_{++}(\mathbb{R})$. Why would function $ f : (X, y) \mapsto y^T X^{-1} y$ be convex? I tried with $(X, x) + t.(Y, y)$ with no result. Also, I thought ...
Orso Forghieri's user avatar
2 votes
0 answers
98 views

Characterization of inverse limits of finite-dimensional convex cones

Consider a countable inverse system $C_1\substack{f_1 \\ \leftarrow} C_2 \substack{f_2 \\ \leftarrow} C_3 \substack{f_3 \\ \leftarrow} \ldots$ where the $C_i$ are finite-dimensional convex cones of ...
postdoc's user avatar
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2 votes
1 answer
169 views

When is a continuous subadditive function (0,1]-superhomogeneous

Continuous version of this Superhomogeneity of subadditive functions Let $f$ be a continuous function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(...
Charles Pehlivanian's user avatar
2 votes
1 answer
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Superhomogeneity of subadditive functions

Let $f$ be a function of $\geq 2$ real variables defined on a convex cone $\mathcal{C}$ in the upper half plane, with $f(0) = 0$. Suppose $f$ is subadditive, i.e. $f(x_1+y_1, \dots, x_n+y_n) \leq f(...
Charles Pehlivanian's user avatar
5 votes
0 answers
238 views

Log-concavity of lattice-functions and convolution

I was looking at the definition of log-concavity: A function $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is said log-concave iff $F(x)\geq 0\forall x\in\mathbb{R}^n$ and $$F(x)^\lambda F(y)^{1-\lambda}\leq ...
Rafael's user avatar
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1 answer
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Worst convex compact set for translational packings of $\mathbb R^d$

A (translational) packing of a convex compact subset (with non-empty interior) $\mathcal C$ of $\mathbb R^d$ is a union of translated non-overlapping (but perhaps touching) copies of $\mathcal C$. The ...
Roland Bacher's user avatar
0 votes
1 answer
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Log-concavity of the modified Bessel function of a second kind

I was searching for some results for the log-concavity of the modified Bessel function of a second type, but I failed. Has there been any known work on this? I am not even sure if it is the modified ...
user808843's user avatar
2 votes
0 answers
121 views

Inscribed square and convexity

Let $b(X)$ be the boundary of any $X$ subset of the plane. Does there exist $A,B$ convex compact sets of the plane, such that $C:=A\setminus B$ is simply connected and not empty, and such that ...
jcdornano's user avatar
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63 views

Length of curves on convex hypersurfaces

Let $\gamma\colon[a,b] \to \mathbb{R}^n$ be a smooth curve. Let $f_i\to f$ be a sequence of convex functions on $\mathbb{R}^n$ converging uniformly on compact subsets to $f$. Let $\hat\gamma(t):=(\...
asv's user avatar
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3 votes
1 answer
134 views

Busemann-Feller lemma in hyperbolic space

The classical Busemann-Feller lemma in Euclidean space says the following. Let $K\subset \mathbb{R}^n$ be a closed convex set. Then for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
asv's user avatar
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How to prove the convexity of a simple function involving a ratio of two polygamma functions?

Let \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\textrm{e}^{-t}\textrm{d}t, \quad \Re(z)>0 \end{equation*} and $$ \psi(z)=[\ln\Gamma(z)]'=\frac{\Gamma'(z)}{\Gamma(z)}. $$ In the literature, ...
qifeng618's user avatar
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Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices

Let $N_{n}=(1/n)_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal. Suppose $n>0$. Let $\delta_{n}$ be the least natural number such that $N_{n}$ can be factored as $N_{n}=A_{...
Joseph Van Name's user avatar
2 votes
1 answer
254 views

Can the subdifferential become unbounded at interior points?

Consider $f: \mathbb{R}^n \to \overline{\mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $\partial f$ be unbounded in the interior of $\text{...
Olórin's user avatar
  • 179
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Biconjugate of a quasiconvex lower semi-continuous function

Let $f:\mathbb{R}^d \to [0,\infty]$ be a quasiconvex lower semi-continuous function whose effective domain $C:=\{x \in \mathbb{R}^d:f(x) < \infty\}$ is nonempty and bounded (and convex since $f$ is ...
Namch96's user avatar
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0 answers
252 views

Convexity of a set of probability densities

Consider the space of probability densities $(P(\mathbb{R}^d), W_2)$ (probability measures on $\mathbb{R}^d$ with 2-Wasserstein distance). How can we determine if a subset $Q$ is convex? I know that a ...
lady gaga's user avatar
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2 votes
0 answers
102 views

How to prove/disprove this surface integral is convex?

This question is related to the following: Convexity of volume in terms of a deformation - the context is summarized below for clarity. In the setting of convex optimization, I am looking for a convex ...
olek n's user avatar
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2 votes
0 answers
136 views

Convexity of volume in terms of a deformation

In the context of convex optimization and mechanics, I am interested in the convexity of the potential energy $U$ of a pressure acting over some volume $V$ enclosed by a surface. Here pressure can be ...
olek n's user avatar
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3 votes
0 answers
52 views

Independence-like property of convex combinations in a vector space

Consider the following property of a set of vectors $S\subset V$, where $V$ is a real vector space: $$ \sum_{i=1}^m w_ix_i = \sum_{i=1}^m u_iy_i,\quad x_i,y_i\in S,\quad 0\le w_i,u_i\le 1, \quad \...
user29985's user avatar
1 vote
0 answers
820 views

The "interior" of a convex set?

Consider any compact metric space $S$ which is a convex subset of a vector space. I am trying to see if the following definition is equivalent/close to some well-known definitions. Consider a convex ...
Lemma1's user avatar
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1 answer
715 views

Simple-looking problem with integrals

Let $f: [0,\infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(0) = 0$. Is it true that if the integral $$ \int_0^{\pi/2} \sin(\theta) f(\lambda \sin(\theta)) \, d\theta $$ is zero ...
alvarezpaiva's user avatar
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2 votes
1 answer
121 views

Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)

This question is related to a previous one. Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
Eduardo Longa's user avatar
-4 votes
1 answer
261 views

strict convexity and Lipschitz continuity [closed]

Consider a continuously differentiable function $f: \mathbb{R}^n \mapsto \mathbb{R}$. If $f$ is strictly convex, does it imply that it is not Lipschitz on $\mathbb{R}^n$? Because if $f$ is strictly ...
Trb2's user avatar
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0 answers
120 views

Relation satisfied by a Gaussian random variable

I want to prove the following relation for $X\sim \mathcal{N}(0,1)$, $x\in \mathbb{R}$ and $f(x)=\mathbb{E}[\max(X,x)]$: $$f(\frac{f(x+1)+f(x-1)}{2})\leq \frac{f(f(x)-1)+f(f(x)+1)}{2}$$ It seems that ...
Pierre's user avatar
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6 votes
0 answers
132 views

Nearby convex set in a nearby space

Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$. Is there a convex set $K'\subset X'$ that is close to $K\subset X$? Two spaces $X$ and $X'$ ...
Anton Petrunin's user avatar
4 votes
1 answer
278 views

Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities: $$ \left<x,u_F\right> \geq -a_F$$ where $u_F\in \...
Luis Ferroni's user avatar
  • 1,879
10 votes
0 answers
211 views

Extremal bases in finite-dimensional Banach spaces

Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the ...
Lviv Scottish Book's user avatar
11 votes
1 answer
212 views

The set of boundary vectors of compact convex body has empty interior

Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$. Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
Taras Banakh's user avatar
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67 votes
3 answers
11k views

Nonconvexity and discretization

Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem. Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
Peter Scholze's user avatar
4 votes
0 answers
63 views

A standard name of a strongly extremal point of a convex set

I need to name somehow points $x$ of a bounded convex set $C$ in a Banach space $X$ such that the set $$\{x^*\in X^*:x^*(x)=\max x^*[C]\}$$ of support functionals at $x$ has non-empty interior in the ...
Taras Banakh's user avatar
  • 40.7k
5 votes
1 answer
204 views

Sufficient condition for geodesic convexity/connectedness

Let $(\Sigma,g)$ be a connected smooth Riemannian manifold without boundary. By a minimizing geodesic I mean a geodesic whose length equals the distance between its endpoints. Let us consider the ...
Ettore Minguzzi's user avatar
1 vote
0 answers
184 views

Stationary distributions of convex combination of stochastic matrices

Consider two irreducible finite state Markov chains with transition matrices $A,B\in\mathbb{R}^{n\times n}$. Let $x$ and $y$ be the unique stationary distributions of $A$ and $B$, respectively. Now ...
jonem's user avatar
  • 179
2 votes
0 answers
51 views

Conjugate of composition in Bochner spaces

Let $H$ be a separable Hilbert space (of non-zero dimension), let $(\Omega,\Sigma,\mu)$ be a finite measure space, and let $L^2(\mu;H)$ be the Bochner-space $\mu$-integrable $H$-valued functions. ...
Catologist_who_flies_on_Monday's user avatar
1 vote
0 answers
268 views

Decomposition of Polyhedral - An example

There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$ \...
holala's user avatar
  • 111
0 votes
1 answer
76 views

Planar function inequality on parallelograms

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ which is convex and satisfies $\frac{\partial{f}^2 }{\partial{x}\partial{y}} \leq 0$. The last condition is ...
Charles Pehlivanian's user avatar
24 votes
4 answers
2k views

Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?

Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. $f:[0,1]\to\mathbb R$ is non-negative, or monotone, or convex; or if it has, say, non-...
Pietro Majer's user avatar
  • 56.3k
2 votes
1 answer
172 views

Sufficient conditions for the convexity of the discrete Fourier transforms

Let $f : [0,2\pi] \to \mathbb{R}$ be some function. Then the discrete Fourier transform of $f$ when sampled at $2\pi i/N$ is then given by $$ X_n := \sum_{i=0}^{N-1}\cos\left(\frac{2\pi n i}{N}\right)...
spaceman's user avatar
  • 575
1 vote
1 answer
281 views

Convexity of discrete Fourier transform

Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
spaceman's user avatar
  • 575
7 votes
2 answers
730 views

Is every face exposed if all extreme points are exposed?

Let $C$ be a non-empty compact convex subset of ${\mathbb R}^d$ such that every extreme point of $C$ is an exposed point of $C$. Does it follow from this that every face of $C$ is an exposed face?
Janko Bracic's user avatar
1 vote
1 answer
215 views

Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$

Working with Slater's inequality (a companion of Jensen's inequality) I found this statement: Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\...
DesmosTutu's user avatar
1 vote
1 answer
38 views

Is this relation between planar convex hulls and heaviest cliques true?

If $P$ is a set of $n$ points in the euclidean plane whose convex hull $\operatorname{CH}(P)$ has $h$ corners, and $Q\subset P$ has $m\le\lfloor\frac{h}{2}\rfloor$ points and maximal sum of pairwise ...
Manfred Weis's user avatar
  • 12.6k
11 votes
0 answers
356 views

Is there yet an example of a non-negative convex polynomial that cannot be written as a sum-of-squares?

I have read that it remains an open question, whether an example can be constructed of a non-negative convex polynomial that cannot be written as a sum-of-squares. My reading includes the following ...
Brian's user avatar
  • 173
9 votes
1 answer
299 views

Closedness of linear image of positive L1 functions

Let $\mathcal X$ be the Banach space of $L^1$ functions on some probability space, $\mathcal Y$ be some other Banach space, $T:\mathcal X\to \mathcal Y$ be some surjective continuous linear map, $\...
e.lipnowski's user avatar
2 votes
0 answers
98 views

Sparse signal recovery (nonlinear case)

Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
Sébastien Loisel's user avatar
5 votes
0 answers
59 views

Self-duality of cones associated with elementary symmetric polynomials

Let $n\ge3$ be an integer, and denote $\sigma_1,\ldots,\sigma_n$ the elementary symmetric polynomials in $n$ indeterminates: $$\sigma_1(X)=X_1+\cdots+X_n,\quad\ldots\quad,\sigma_n(X)=X_1\cdots X_n.$$ ...
Denis Serre's user avatar
  • 51.5k
0 votes
0 answers
99 views

About the definition of lineal convexity

I have been trying to understand the definition of lineal convexity. I am reading the article Duality of functions defined in lineally convex sets by Christer O. Kiselman. For a set $A\subset \mathbb{...
user429197's user avatar

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